Power Play (Exponents) | FIO

Question 2

Express each of the following as a product of powers of their prime factors in exponential form:

(i) 648

(ii) 405

(iii) 540

(iv) 3600

Solution

Understand the Question

To express a number as a product of powers of its prime factors in exponential form:
- Perform prime factorisation by repeatedly dividing the number by prime numbers ( $2, 3, 5, 7, \dots$ ) until the quotient is $1$ .
- Count how many times each prime factor appears and write it using exponents ( $a \times a \times \dots \times a = a^n$ ).

(i) 648

Step 1 · Prime Factorisation of 648

Diagram 1

\begin{aligned} 648 &= 2 \times 324 \\ &= 2 \times 2 \times 162 \\ &= 2 \times 2 \times 2 \times 81 \\ &= 2 \times 2 \times 2 \times 3 \times 27 \\ &= 2 \times 2 \times 2 \times 3 \times 3 \times 9 \\ &= 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \\ &= 2^3 \times 3^4 \end{aligned}

Answer

(i) $2^3 \times 3^4$

(ii) 405

Step 1 · Prime Factorisation of 405

\begin{aligned} 405 &= 3 \times 135 \\ &= 3 \times 3 \times 45 \\ &= 3 \times 3 \times 3 \times 15 \\ &= 3 \times 3 \times 3 \times 3 \times 5 \\ &= 3^4 \times 5 \end{aligned}

Answer

(ii) $3^4 \times 5$

(iii) 540

Step 1 · Prime Factorisation of 540

\begin{aligned} 540 &= 2 \times 270 \\ &= 2 \times 2 \times 135 \\ &= 2 \times 2 \times 3 \times 45 \\ &= 2 \times 2 \times 3 \times 3 \times 15 \\ &= 2 \times 2 \times 3 \times 3 \times 3 \times 5 \\ &= 2^2 \times 3^3 \times 5 \end{aligned}

Answer

(iii) $2^2 \times 3^3 \times 5$

(iv) 3600

Step 1 · Prime Factorisation of 3600

\begin{aligned} 3600 &= 2 \times 1800 \\ &= 2 \times 2 \times 900 \\ &= 2 \times 2 \times 2 \times 450 \\ &= 2 \times 2 \times 2 \times 2 \times 225 \\ &= 2 \times 2 \times 2 \times 2 \times 3 \times 75 \\ &= 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 25 \\ &= 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 5 \\ &= 2^4 \times 3^2 \times 5^2 \end{aligned}

Answer

(iv) $2^4 \times 3^2 \times 5^2$

Common Mistakes

Using Composite Numbers: Dividing by composite numbers like $4, 6, 8, \text{ or } 9$ instead of strictly using prime numbers ( $2, 3, 5, 7, \dots$ ).
Miscounting Exponents: Missing or double counting a factor during repeated division, leading to an incorrect power.

More questions in FIO

Express the following in exponential form:

(i) $6 \times 6 \times 6 \times 6$

(ii) $y \times y$

(iii) $b \times b \times b \times b$

(iv) $5 \times 5 \times 7 \times 7 \times 7$

(v) $2 \times 2 \times a \times a$

(vi) $a \times a \times a \times c \times c \times c \times c \times d$

Express each of the following as a product of powers of their prime factors in exponential form:

(i) 648

(ii) 405

(iii) 540

(iv) 3600

Write the numerical value of each of the following:

(i) $2 \times 10^3$

(ii) $7^2 \times 2^3$

(iii) $3 \times 4^4$

(iv) $(-3)^2 \times (-5)^2$

(v) $3^2 \times 10^4$

(vi) $(-2)^5 \times (-10)^6$

Find out the units digit in the value of $2^{224} \div 4^{32}$ ? [Hint: $4 = 2^2$ ]

There are 5 bottles in a container. Every day, a new container is brought in. How many bottles would there be after 40 days?

Write the given number as the product of two or more powers in three different ways. The powers can be any integers.

(i) $64^3$ (ii) $192^8$ (iii) $32^{-5}$

Examine each statement below and find out if it is 'Always True', 'Only Sometimes True', or 'Never True'. Explain your reasoning.

(i) Cube numbers are also square numbers.

(ii) Fourth powers are also square numbers.

(iii) The fifth power of a number is divisible by the cube of that number.

(iv) The product of two cube numbers is a cube number.

(v) $q^{46}$ is both a 4th power and a 6th power ( $q$ is a prime number).

Simplify and write these in the exponential form.

(i) $10^{-2} \times 10^{-5}$

(ii) $5^7 \div 5^4$

(iii) $9^{-7} \div 9^4$

(iv) $(13^{-2})^{-3}$

(v) $m^5 n^{12} (mn)^9$

If $12^2 = 144$ , what is

(i) $(1.2)^2$

(ii) $(0.12)^2$

(iii) $(0.012)^2$

(iv) $120^2$

Q10

Circle the numbers that are the same—

$2^4 \times 3^6 \qquad 6^4 \times 3^2 \qquad 6^{10} \qquad 18^2 \times 6^2 \qquad 6^{24}$

Q11

Identify the greater number in each of the following—

(i) $4^3$ or $3^4$

(ii) $2^8$ or $8^2$

(iii) $100^2$ or $2^{100}$

Q12

A dairy plans to produce 8.5 billion packets of milk in a year. They want a unique ID (identifier) code for each packet. If they choose to use the digits $0$ – $9$ , how many digits should the code consist of?

Q13

64 is a square number ( $8^2$ ) and a cube number ( $4^3$ ). Are there other numbers that are both squares and cubes? Is there a way to describe such numbers in general?

Q14

A digital locker has an alphanumeric (it can have both digits and letters) passcode of length 5. Some example codes are G89P0, 38098, BRJKW, and 003AZ. How many such codes are possible?

Q15

The worldwide population of sheep (2024) is about $10^9$ , and that of goats is also about the same. What is the total population of sheep and goats?

(i) $20^9$

(ii) $10^{11}$

(iii) $10^{10}$

(iv) $10^{18}$

(v) $2 \times 10^9$

(vi) $10^9 + 10^9$

Q16

Calculate and write the answer in scientific notation:

(i) If each person in the world had 30 pieces of clothing, find the total number of pieces of clothing.

(ii) There are about 100 million bee colonies in the world. Find the number of honeybees if each colony has about 50,000 bees.

(iii) The human body has about 38 trillion bacterial cells. Find the bacterial population residing in all humans in the world.

(iv) Total time spent eating in a lifetime in seconds.

Q17

What was the date 1 arab/1 billion seconds ago?